I think this is a pretty good intuitive introduction.

I think there’s a bit of muddling between the fundamental group and first homology group with the article: the algebra of the fundamental group where loops are merged together cares about the order you do each loop and I don’t think it’s intuitive that a + b = b + a for the torus, indeed I found this quite unintuitive when I learned it. Note that the group isn’t always commutative. On the other hand the elements in the homologous group are, to some extent, superpositions of countable many loops and so the group will always be commutative (as far as I know). The direction of those loops matters but the order does not. That said, I still think the article is ok for the level it is aimed at.

Another complaint is that I feel like it can be read in this weird way where topology is all about defining what precisely a hole is. Instead I would say that the things here (Betti numbers, generators of the fundamental group, etc) are merely tools which topologies have found to be convenient in topology and which have been called holes as they roughly correspond to the colloquial meaning of the word.

what follows is a motivation but not a history. It is also biased by the way I learned topology.

For an example, a nice early question in topology is: Why is R^1, the line, different from R^2, the plane. Here different means “not homeomorphic” which has a slightly involved definition. The answer comes in two parts: firstly that if you remove a point from a line you split it in two while that is not true of the plane, and secondly is that connectedness (whether something is split in two) is invariant under homeomorphism (and some boring things about the definition of a homeomorphism). Therefore the two spaces cannot be homemorphic.

Here we have, in some sense used the nature of a hole to answer a question about topology. But here are two questions that it doesn’t answer:

1. Are R^2 and R^3 different?

2. Are R^2, the plane, and S^2, the surface of a sphere, different?

3. If you remove a point from R^2, is that different from removing two points?

Removing single or multiple points doesn’t really help in these. And dealing with lines leads to problems (who’s to say that removing a line from R^2 won’t correspond to removing a plane from R^3, breaking the logic above? The sets of points being removed are uncountably infinite in both cases.) The solution to the first two problems is simply-connectedness: a space is simply connected if you can find a line between any two points and you can, in some formal sense, slide any line between those points into any other line between those two points. R^2 and S^2 are simply connected. But if you remove a point from R^2 it is not simply connected as the line that does a loop around the missing point cannot be slid around into a line that doesn’t do the loop. If you remove a point from S^2 it is still simply connected (if you have a loop around the missing point just slide it to the other side of the sphere where it can be unraveled). The same argument works for the other problem as if you remove a point from R^3 it is still simply connected.

For the last problem these ideas are not sufficient. But following similar techniques leads to the definition of the fundamental group (roughly the algebra of adding up loops discussed in the article) which can solve that problem. The next problem to solve is to show that R^3 and R^4 are different. This can motivate homology (maybe you could get away with removing a line to make R^3 not simply connected but that sounds hard. And it won’t help much with the general case of showing that R^n and R^m are different if n and m are.)

In these cases there are things that might be talked about as holes being measured but the goal isn’t to define what a hole is but rather to answer questions about topology.